2015, 2015(special): 1034-1040. doi: 10.3934/proc.2015.1034

Recurrence of multi-dimensional diffusion processes in Brownian environments

1. 

Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501, Japan

2. 

Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Citation: Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034
References:
[1]

T. Brox, A one-dimensional diffusion process in a Wiener medium., Ann. Probab., 14 (1986), 1206.   Google Scholar

[2]

M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization,, in, 1250 (1987), 87.   Google Scholar

[3]

K. Ichihara, Some global properties of symmetric diffusion processes., Publ. RIMS, 14 (1978), 441.   Google Scholar

[4]

K. Itô, On the ergodicity of a certain stationary process., Proc. Imp. Acad., 20 (1944), 54.   Google Scholar

[5]

K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'', Springer-Verlag, (1965).   Google Scholar

[6]

D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments,, J. Korean Math. Soc., 48 (2011), 147.   Google Scholar

[7]

G. Maruyama, The harmonic analysis of stationary stochastic processes,, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45.   Google Scholar

[8]

P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment,, Probab. Theory Related Fields, 99 (1994), 549.   Google Scholar

[9]

Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment,, Theory Probab. Appl., 27 (1982), 256.   Google Scholar

[10]

F. Solomon, Random walks in a random environment,, Ann. Probab., 3 (1975), 1.   Google Scholar

[11]

H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments., Statist. Proba. Lett., 69 (2004), 171.   Google Scholar

[12]

H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments,, in, 9 (1987), 189.   Google Scholar

[13]

H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment,, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377.   Google Scholar

show all references

References:
[1]

T. Brox, A one-dimensional diffusion process in a Wiener medium., Ann. Probab., 14 (1986), 1206.   Google Scholar

[2]

M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization,, in, 1250 (1987), 87.   Google Scholar

[3]

K. Ichihara, Some global properties of symmetric diffusion processes., Publ. RIMS, 14 (1978), 441.   Google Scholar

[4]

K. Itô, On the ergodicity of a certain stationary process., Proc. Imp. Acad., 20 (1944), 54.   Google Scholar

[5]

K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'', Springer-Verlag, (1965).   Google Scholar

[6]

D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments,, J. Korean Math. Soc., 48 (2011), 147.   Google Scholar

[7]

G. Maruyama, The harmonic analysis of stationary stochastic processes,, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45.   Google Scholar

[8]

P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment,, Probab. Theory Related Fields, 99 (1994), 549.   Google Scholar

[9]

Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment,, Theory Probab. Appl., 27 (1982), 256.   Google Scholar

[10]

F. Solomon, Random walks in a random environment,, Ann. Probab., 3 (1975), 1.   Google Scholar

[11]

H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments., Statist. Proba. Lett., 69 (2004), 171.   Google Scholar

[12]

H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments,, in, 9 (1987), 189.   Google Scholar

[13]

H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment,, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377.   Google Scholar

[1]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[2]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[3]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[4]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[5]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[6]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103

[7]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[8]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[9]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[10]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[11]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[12]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[13]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

 Impact Factor: 

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]